1. Field of the Invention
The present invention relates generally to marine seismic surveying and, more particularly, to a method for attenuating the effect of surface multiples in a marine seismic signal.
2. Description of the Related Art
Seismic surveying is a method for determining the structure of subterranean formations in the earth. Seismic surveying typically utilizes seismic energy sources which generate seismic waves and seismic receivers which are strategically positioned to detect the seismic waves. The seismic waves propagate into the formations in the earth, where a portion of the waves reflects from interfaces between subterranean formations. The amplitude and polarity of the reflected waves are determined by the differences in acoustic impedance between the rock layers comprising the subterranean formations. The acoustic impedance of a rock layer is the product of the acoustic propagation velocity within the layer and the density of the layer. The reflected seismic waves are detected by the seismic receivers, which convert the reflected waves into representative electrical signals. The signals are typically transmitted by electrical, optical, radio or other means to devices which record the signals. Through analysis of the recorded signals, the shape, position and composition of subterranean formations can be determined.
Land seismic surveying is a method for determining the structure of subterranean formations beneath the surface of the earth. The seismic energy source typically used is an apparatus capable of delivering a series of impacts or mechanical vibrations to the surface of the earth or the detonation of an explosive charge near the surface of the earth. The seismic receiver typically used in seismic surveying on land is a motion sensor, such as a geophone or an accelerometer. The seismic sources and seismic receivers are typically placed on the surface of the earth, although either source or receiver may be placed in a borehole for vertical seismic profiling. Both the seismic sources and the seismic receivers are typically repositioned to cover the survey area.
Marine seismic surveying is a method for determining the structure of subterranean formations underlying bodies of water. Marine seismic surveying typically utilizes seismic energy sources and seismic receivers located in the water which are either towed behind a vessel or positioned on the water bottom from a vessel. The energy source is typically an explosive device or compressed air system which generates seismic energy, which then propagates as seismic waves through the body of water and into the earth formations below the bottom of the water. As the seismic waves strike interfaces between subterranean formations, a portion of the seismic waves reflects back through the earth and water to the seismic receivers, to be detected, transmitted, and recorded. The seismic receivers typically used in marine seismic surveying are pressure sensors, such as hydrophones. Both the sources and receivers may be repositioned to cover the survey area.
Seismic waves, however, do not reflect only from the interfaces between subterranean formations, as would be desired. Seismic waves also reflect from the water bottom and the water surface, and the resulting reflected waves themselves continue to reflect. Waves which reflect multiple times are called "multiples". Waves which reflect multiple times in the water layer between the water surface above and the water bottom below are called "water-bottom multiples". Water-bottom multiples have long been recognized as a problem in marine seismic processing and interpretation, so multiple attenuation methods based on the wave equation have been developed to handle water-bottom multiples. However, a larger set of multiples containing water-bottom multiples as a subset can be defined. The larger set includes multiples with lower reflections from subterranean formations in addition to reflections from the water bottom. The multiples in the larger set have in common their downward reflection at the water surface and thus are called "surface multiples". FIG. 1, discussed below, provides examples of different types of reflections.
FIG. 1 shows a diagrammatic view of marine seismic surveying. The procedure is designated generally as 100. Subterranean formations to be explored, such as 102 and 104, lie below a body of water 106. Seismic energy sources 108 and seismic receivers 110 are positioned in the body of water 106, typically by one or more seismic vessels (not shown). A seismic source 108, such as an air gun, creates seismic waves in the body of water 106 and a portion of the seismic waves travels downward through the water toward the subterranean formations 102 and 104 beneath the body of water 106. When the seismic waves reach a seismic reflector, a portion of the seismic waves reflects upward and a portion of the seismic waves continues downward. The seismic reflector can be the water bottom 112 or one of the interfaces between subterranean formation, such as interface 114 between formations 102 and 104. When the reflected waves travelling upward reach the water/air interface at the water surface 116, a majority portion of the waves reflects downward. Continuing in this fashion, seismic waves can reflect multiple times between the water bottom 112 or formation interfaces below and the water surface 116 above, as described more fully below. Each time the reflected waves propagate past the position of a seismic receiver 110, the receiver 110 senses the reflected waves and generates representative signals.
Primary reflections are those seismic waves which have reflected only once, from the water bottom 112 or an interface between subterranean formations, before being detected by a seismic receiver 110. An example of a primary reflection is shown in FIG. 1 by raypaths 120 and 122. Seismic waves are generated by a seismic source 108 and a portion of the waves travels downward through the body of water 106 and into the subterranean formation 102 along raypath 120. A portion of the seismic waves reflects from the interface 114 between formations 102 and 104. The reflected waves travel upward through the formation 102 and back into the body of water 106 along raypath 122. The reflected waves travel past a receiver 110, which detects the waves and generates a representative signal. Primary reflections contain the desired information about the subterranean formations which marine seismic surveying seeks. Surface multiples are those waves which have reflected multiple times between the water surface 116 and lower reflectors, such as the water bottom 112 or formation interfaces, before being sensed by a receiver 110. An example of a surface multiple which is specifically a water bottom multiple is shown starting at raypath 130. Seismic waves are generated by a seismic source 108 and a portion of the waves travels downward through the body of water 106 along raypath 130. A portion of the seismic waves reflects from the water bottom 112 and travels back upward through the body of water 106 along raypath 132. A portion of the reflected waves reflects from the water surface 116 and travels back downward through the body of water 106 along raypath 134. A portion of the twice-reflected waves reflects again from the water bottom 112 and travels back upward through the body of water 106 along raypath 136. The thrice-reflected waves travel past a receiver 110, which detects the waves and generates a representative signal. The surface multiple starting at raypath 130 is a multiple of order one, since the multiple contains one reflection from the water surface 116.
Two examples of general surface multiples which reflect from both the water bottom 112 and formation interfaces are shown starting at raypath 140 and starting at raypath 160. In the first example, seismic waves travel downward from a seismic source 108 through the body of water 106 along raypath 140, reflect from the water bottom 112 a first time and travel upward through the water 106 along raypath 142, reflect from the water surface 116 a first time and travel downward through the water 106 and then the formation 102 along raypath 144, reflect from the interface 114 and travel upward through the formation 102 and then the water 106 along raypath 146, reflect from the water surface 116 a second time and travel downward through the water 106 along raypath 148, and finally reflect from the water bottom 112 a second time before traveling upward through the formation 102, the water 106 and past a receiver 110 along raypath 150. In the second example, the seismic waves travel downward through the water 106 from a seismic source 108 along raypath 160, reflect from the water bottom 112 a first time and travel upward through the water 106 along raypath 162, reflect from the water surface 116 a first time and travel downward through the water 106 along raypath 164, reflect from the water bottom 112 a second time and travel upward through the water 106 along raypath 166, reflect from the water surface 116 a second time and travel downward through the water 106 and then the formation 102 along raypath 168, and finally reflect from the interface 114 before traveling upward through the formation 102, the water 106 and past a receiver 110 along raypath 170. Both of these latter two examples of surface multiples are multiples of order two, since the multiples contain two reflections from the water surface 116. In general, a surface multiple is of order i if the multiple contains i reflections from the water surface 116. Surface multiples are extraneous noise which obscures the desired primary reflection signal.
Surface multiple attenuation is a prestack inversion of a recorded wavefield which removes all orders of all surface multiples present within the marine seismic signal. Unlike some wave-equation-based multiple-attenuation algorithms, surface multiple attenuation does not require any modeling of or assumptions regarding the positions, shapes and reflection coefficients of the multiple-causing reflectors. Instead, surface multiple attenuation relies on the internal physical consistency between primary and multiple events that must exist in any properly recorded marine data set. The information needed for the surface multiple attenuation process is already contained within the seismic data.
In the following discussion, let upper-case letters represent the original seismic wavefields, the corresponding recorded data sets, or the corresponding data cubes or matrices. Thus let D represent a marine seismic data set corresponding to a wavefield D. The wavefield D can be divided into two parts, EQU D=P+M. (1)
The primary wavefield, P, represents that portion of D which contains no surface multiples. The surface multiples wavefield, M, represents that portion of D which contains surface multiples of any order. Surface multiple attenuation is a processing method for removing the multiples wavefield M from the recorded wavefield D to yield the desired primary wavefield P.
For each i from 1 to .infin., let M.sub.i represent that portion of M containing surface multiples of order i. Then the surface multiple wavefield M can be further decomposed into an infinite sum of different orders, EQU M=M.sub.1 +M.sub.2 + . . . +M.sub.i + . . . . (2)
Recorded data sets have a finite duration, so only a finite number of terms from Eq. (2) are needed to represent the corresponding wavefield. Substituting an appropriately truncated Eq. (2) into Eq. (1) yields EQU D=P+M.sub.1 +M.sub.2 + . . . +M.sub.n, (3)
for some value n.
The process of surface multiple attenuation assumes that surface multiple events M.sub.i of order i can be predicted from knowledge of both the surface multiple events M.sub.i-1 of order i-1 and the primary wavefield P. This assumption means that there exists some mathematical operator O such that EQU M.sub.i =POM.sub.i-1. (4)
Inserting Eq. (4) into Eq. (3) and factoring out first P and then O yields ##EQU1## Define a truncated version of D by ##EQU2## In practice, as will be discussed later, D.sub.T would be approximated by truncating the traces in D in time rather than actually constructing and subtracting M.sub.n from D. Inserting Eq. (6) into Eq. (5) yields the compact form EQU D=P[1+OD.sub.T ]. (7)
Eq. (7) is a formula for recursive forward modeling of surface multiples. Eq. (7) represents adding the events of order n to the wavefield containing all events up to and including order n-1. If the bracketed expression in Eq. (7) has an inverse, then Eq. (7) can be inverted to yield EQU P=D[1+OD.sub.T ].sup.-1. (8)
Eq. (8) is the inverse of the recursive forward modeling equation, Eq. (7). Eq. (8) states that if a suitable operator O can be found, then the primary wavefield P, free of surface multiples, can be computed directly from the recorded wavefield D. The operator O being suitable means that the operator O must be both geophysically and mathematically plausible. The operator O being geophysically plausible means that the operator O satisfies Eq. (4). The operator O being mathematically plausible means firstly that the factorizations in Eq. (5) are valid and secondly that the inverse of the bracketed expression in Eq. (7) exists and thus Eq. (8) is valid.
Let lower-case letters represent individual traces or events within the wavefields or data sets. Thus m.sub.i is a multiple event of order i within a trace d in the wavefield D. Riley and Claerbout, "2-D Multiple Reflections" Geophysics, vol 41, 1976, pp. 592-620, derive the one-dimensional versions of Eqs. (7) and (8). Assume that the earth has a single, flat, reflecting layer, is laterally homogeneous and the marine source creates a spike-like plane wave traveling vertically downward. Under these conditions, every trace in the wavefield D is the same, so the entire wavefield D can be represented by a single trace d. Letting the reflection coefficient of the water surface be -1, the following recursive formula holds for predicting the surface multiple event m.sub.i of order i from the surface multiple event m.sub.i-1 of order i-1 and the primary event p in the trace d, EQU m.sub.i =-p*m.sub.i-1, (9)
where * represents convolution in the time domain. Eq. (9) is a one-dimensional version of Eq. (4). Here, the operator O has become convolution followed by multiplication by -1. The factorizations in Eq. (5) are mathematically valid since convolution is a commutative process. The inversion in Eq. (8) is mathematically valid since it is simply deconvolution. In this case, the one-dimensional version of Eq. (8) becomes EQU p=d*[1-d.sub.T ].sup.-1. (10)
The expression in brackets in Eq. (10) acts as a filter whose inverse deconvolves the surface multiples in trace d. Eq. (10) works equally well for one-dimensional models of surface multiple attenuation having any number of reflective layers.
Directly applying Eqs. (9) and (10) to the two- or three-dimensional cases of surface multiple attenuation is not practical, however. A two- or three-dimensional version of Eq. (9) must honor the wave equation. The Kirchhoff integral, a mathematical statement of Huygens' principle, does honor the wave equation. The Kirchhoff integral provides a two- or three-dimensional generalization of Eq. (9), and, thus, is the basis of a geophysically suitable operator O. Several different approaches to two-dimensional surface multiple attenuation are described in the literature. Riley and Claerbout, "2-D Multiple Reflections", Geophysics, vol. 41, 1976, pp. 592-620, extend their one-dimensional deconvolution to two-dimensional using a finite difference scheme based on the scalar wave equation and using information on the source wavelet and reflectivities. Fokkema and Van den Berg, "Removal of Surface-Related Wave Phenomena: the Marine Case" 60th Annual International Meeting, SEG, Expanded Abstracts, 1990, pp. 1689-1692, describe a method of removing surface multiples that is derived from the Rayleigh reciprocity theorem. Solution is by direct matrix inversion or an iterative Neumann series, using information about the source wavelet and water layer properties. Verschuur, "Surface-Related Multiple Elimination in Terms of Huygens' Sources", J. of Seismic Exploration, vol. 1, 1992, pp. 49-59, and Verschuur et al., "Adaptive Surface-Related Multiple Elimination" Geophysics, vol. 57, no. 1, 1992, pp. 1166-1177, solve the surface multiple problem using an f-x domain method based on Huygens' principle, using information about the source wavelet and free surface reflectivity properties. A scaled estimate of the source wavelet may be adaptively calculated Carvalho et al., "Examples of a Nonlinear Inversion Method Based on the T Matrix of Scattering Theory: Application to Multiple Suppression", 61st Annual International Meeting, SEG, Expanded Abstracts, 1991, pp. 1319-1322, formulate a method of multiple suppression in terms of T-matrix scattering theory, using an estimate of the source wavelet. All of these methods are closely related because all of them must honor the acoustic wave equation.